ltexpri - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ltexpri		GIF version

Theorem ltexpri 7697

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

**Assertion**
Ref	Expression
ltexpri	⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem ltexpri Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
2	1	eleq1d 2265	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (2^nd ‘𝐴) ↔ 𝑣 ∈ (2^nd ‘𝐴)))
3		simpl 109	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑦 = 𝑢)
4	1, 3	oveq12d 5943	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 +_Q 𝑦) = (𝑣 +_Q 𝑢))
5	4	eleq1d 2265	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵)))
6	2, 5	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ (𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
7	6	cbvexdva 1944	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
8	7	cbvrabv 2762	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}
9	1	eleq1d 2265	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (1^st ‘𝐴) ↔ 𝑣 ∈ (1^st ‘𝐴)))
10	4	eleq1d 2265	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵)))
11	9, 10	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ (𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
12	11	cbvexdva 1944	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
13	12	cbvrabv 2762	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}
14	8, 13	opeq12i 3814	. . 3 ⊢ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ = ⟨{𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}, {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}⟩
15	14	ltexprlempr 7692	. 2 ⊢ (𝐴<_P 𝐵 → ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P)
16	14	ltexprlemfl 7693	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (1^st ‘𝐵))
17	14	ltexprlemrl 7694	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐵) ⊆ (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
18	16, 17	eqssd 3201	. . 3 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵))
19	14	ltexprlemfu 7695	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (2^nd ‘𝐵))
20	14	ltexprlemru 7696	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘𝐵) ⊆ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
21	19, 20	eqssd 3201	. . 3 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))
22		ltrelpr 7589	. . . . . . 7 ⊢ <_P ⊆ (P × P)
23	22	brel 4716	. . . . . 6 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
24	23	simpld 112	. . . . 5 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
25		addclpr 7621	. . . . 5 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P) → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
26	24, 15, 25	syl2anc 411	. . . 4 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
27	23	simprd 114	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
28		preqlu 7556	. . . 4 ⊢ (((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
29	26, 27, 28	syl2anc 411	. . 3 ⊢ (𝐴<_P 𝐵 → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
30	18, 21, 29	mpbir2and 946	. 2 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵)
31		oveq2 5933	. . . 4 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → (𝐴 +_P 𝑥) = (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩))
32	31	eqeq1d 2205	. . 3 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → ((𝐴 +_P 𝑥) = 𝐵 ↔ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵))
33	32	rspcev 2868	. 2 ⊢ ((⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P ∧ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵) → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)
34	15, 30, 33	syl2anc 411	1 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ∃wrex 2476 {crab 2479 ⟨cop 3626 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 1^st c1st 6205 2^nd c2nd 6206 Qcnq 7364 +_Q cplq 7366 Pcnp 7375 +_P cpp 7377 <_P cltp 7379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-eprel 4325 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-1o 6483 df-2o 6484 df-oadd 6487 df-omul 6488 df-er 6601 df-ec 6603 df-qs 6607 df-ni 7388 df-pli 7389 df-mi 7390 df-lti 7391 df-plpq 7428 df-mpq 7429 df-enq 7431 df-nqqs 7432 df-plqqs 7433 df-mqqs 7434 df-1nqqs 7435 df-rq 7436 df-ltnqqs 7437 df-enq0 7508 df-nq0 7509 df-0nq0 7510 df-plq0 7511 df-mq0 7512 df-inp 7550 df-iplp 7552 df-iltp 7554

This theorem is referenced by: lteupri 7701 ltaprlem 7702 ltaprg 7703 ltmprr 7726 recexgt0sr 7857 mulgt0sr 7862 map2psrprg 7889

W3C validator